Open access peer-reviewed chapter

Boundary-Layer Flow in a Porous Medium of a Nanofluid Past a Vertical Cone

By F.M. Hady, F.S. Ibrahim, S.M. Abdel-Gaied and M.R. Eid

Submitted: November 30th 2011Reviewed: September 17th 2012Published: October 31st 2012

DOI: 10.5772/53497

Downloaded: 3679

1. Introduction

The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.

The derivation of the empirical equations which govern the flow and heat transfer in a porous medium has been discussed in [1-5]. The natural convection on vertical surfaces in porous media has been studied used Darcy’s law by a number of authors [6–20]. Boundary layer analysis of natural convection over a cone has been investigated by Yih [21-24]. Murthy and Singh [25] obtained the similarity solution for non-Darcy mixed convection about an isothermal vertical cone with fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity, but a semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained Roy and Anilkumar [26]. The laminar steady nonsimilar natural convection flow of gases over an isothermal vertical cone has been investigated by Takhar et al. [27]. The development of unsteady mixed convection flow of an incompressible laminar viscous fluid over a vertical cone has been investigated by Singh and Roy [28] when the fluid in the external stream is set into motion impulsively, and at the same time the surface temperature is suddenly changed from its ambient temperature. An analysis has been carried out by Kumari and Nath [29] to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection. Cheng [30-34] focused on the problem of natural convection from a vertical cone in a porous medium with mixed thermal boundary conditions, Soret and Dufour effects and with variable viscosity.

The conventional heat transfer fluids including oil, water and ethylene glycol etc. are poor heat transfer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years. Choi [35] introduced the term nanofluid refers to these kinds of fluids by suspending nanoparticles in the base fluid. Khanafer et al. [36] investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The convective boundary-layer flow over vertical plate, stretching sheet and moving surface studied by numerous studies and in the review papers Buongiorno [37], Daungthongsuk and Wongwises [38], Oztop [39], Nield and Kuznetsov [40,41], Ahmad and Pop [42], Khan and Pop [43], Kuznetsov and Nield [44,45] and Bachok et al. [46].

From literature survey the base aim of this work is to study the free convection boundary-layer flow past a vertical cone embedded in a porous medium filled with a nanofluid, the basic fluid being a non-Newtonian fluid by using similarity transformations. The reduced coupled ordinary differential equations are solved numerically. The effects of the parameters governing the problem are studied and discussed.


2. Mathematical formulation of the problem

Consider the problem of natural convection about a downward-pointing vertical cone of half angle γembedded in a porous medium saturated with a non-Newtonian power-law nanofluid. The origin of the coordinate system is placed at the vertex of the full cone, with xbeing the coordinate along the surface of the cone measured from the origin and ybeing the coordinate perpendicular to the conical surface Fig (1). The temperature of the porous medium on the surface of the cone is kept at constant temperatureTw, and the ambient porous medium temperature is held at constant temperatureT.

Figure 1.

A schematic diagram of the physical model.

The nanofluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermo physical properties of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [39]). Assuming that the thermal boundary layer is sufficiently thin compared with the local radius, the equations governing the problem of Darcy flow through a homogeneous porous medium saturated with power-law nanofluid near the vertical cone can be written in two-dimensional Cartesian coordinates (x,y)as:


Where uand vare the volume-averaged velocity components in the xand ydirections, respectively, Tis the volume-averaged temperature. nis the power-law viscosity index of the power-law nanofluid and gis the gravitational acceleration. m=γ=0corresponds to flow over a vertical flat plate and m=1corresponds to flow over a vertical cone. nis the viscosity index. For the case ofn=1, the base fluid is Newtonian. We note that n<1and n>1represent pseudo-plastic fluid and dilatant fluid, respectively. Property ρnfand μnfare the density and effective viscosity of the nanofluid, and Kis the modified permeability of the porous medium. Furthermore, αnfand βnfare the equivalent thermal diffusivity and the thermal expansion coefficient of the saturated porous medium, which are defined as (see Khanafer et al. [36]):


Here ϕis the solid volume fraction.

The associated boundary conditions of Eqs. (1)-(3) can be written as:

v=0;T=Twat  y=0;u=0;TTas  y,E5

where μfis the viscosity of the basic fluid, ρfand ρsare the densities of the pure fluid and nanoparticle, respectively, (ρCp)fand (ρCp)sare the specific heat parameters of the base fluid and nanoparticle, respectively, kfand ksare the thermal conductivities of the base fluid and nanoparticle, respectively. The local radius to a point in the boundary layer rcan be represented by the local radius of the vertical coner=xsinγ.

Pure water997.141790.61321
Copper (Cu)89333854011.67
Silver (Ag)105002354291.89
Alumina (Al2O3)3970765400.85
Titanium Oxide (TiO2)4250686.28.95380.9

Table 1.

Thermo-physical properties of water and nanoparticles [39].

By introducing the following non-dimensional variables:

η=yxRax1l2,          f(η)=ψ(x,y)αfrmRax1l2,θ(η)=TTTwT.E6

The continuity equation is automatically satisfied by defining a stream function ψ(x,y)such that:

rmu=ψy and rmv=ψx.E7



Integration the momentum Eq. (2) we have:


Substituting variables (6) into Eqs. (1)–(5) with Eq. (9), we obtain the following system of ordinary differential equations:


along with the boundary conditions:


where primes denote differentiation with respect toη, the quantity of practical interest, in this chapter is the Nusselt number Nuxwhich is defined in the form:


where hdenotes the local heat transfer coefficient.

3. Results and discussion

In this study we have presented similarity reductions for the effect of a nanoparticle volume fraction on the free convection flow of nanofluids over a vertical cone via similarity transformations. The numerical solutions of the resulted similarity reductions are obtained for the original variables which are shown in Eqs. (10) and (11) along with the boundary conditions (12) by using the implicit finite-difference method. The physical quantity of interest here is the Nusselt number Nuxand it is obtained and shown in Eqs. (13) and (14). The distributions of the velocityf(η), the temperature θ(η)from Eqs.(10) and (11) and the Nusselt number in the case of Cu-water and Ag-water are shown in Figs. 2–8. The computations are carried for various values of the nanoparticles volume fraction for different types of nanoparticles, when the base fluid is water. Nanoparticles volume fraction ϕis varied from 0 to 0.3. The nanoparticles used in the study are from Copper (Cu), Silver (Ag), Alumina (Al2O3) and Titanium oxide (TiO2).

In order to verify the accuracy of the present method, we have compared our results with those of Yih [22] for the rate of heat transfer θ(0)in the absence of the nanoparticles(ϕ=0). The comparisons in all the above cases are found to be in excellent agreement, as shown in Table 2. It is clear that as a geometry shape parameter mincreases, the local Nusselt number increases. While Table 3 depict the heat transfer rate θ(0)for various values of nanoparticles volume fraction ϕfor different types of nanoparticles when the base fluid is water. Figs. 2 and 3 show the effects of the nanoparticle volume fraction φon the velocity distribution in the case of Cu-water whenϕ=0,0.05,0.1,0.15,0.2,0.3. It is noted that the velocity along the cone increases with the nanoparticle volume fraction in both of the two cases (i.e. Cu-water and Ag-water), moreover the velocity distribution in the case of Ag-water is larger than that for Cu-water. We can show that the change of the velocity distribution when we use different types of nanoparticles from Fig. 4, which depict the Ag-nanoparticles are the highest when the base fluid is water and whenϕ=0.1. Thus the presence of the nanoparticles volume fraction increases the momentum boundary layer thickness.

Figure 2.

Effects of the nanoparticle volume fractionϕon velocity distributionf(η)in the case of Cu-Water.

nVertical plateVertical cone
Yih [22]Present methodYih [22]Present method

Table 2.

Comparison of results for the reduced Nusselt number θ(0)for vertical plate (λ=0)and vertical cone (λ=1)whenϕ=0.


Table 3.

Values of θ(0)for various values of ϕwhenn=1.

Figure 3.

Effects of the nanoparticle volume fractionϕon velocity distributionf(η)in the case of Ag-Water.

Figure 4.

Velocity profilesf(η)for different types of nanofluids whenϕ=0.1.

Figs. 5 and 6 are presented to show the effect of the volume fraction of nanoparticles Cu and Ag respectively, on temperature distribution. These figures illustrate the streamline for different values ofϕ, when the volume fraction of the nanoparticles increases from 0 to 0.3, the thermal boundary layer is increased. This agrees with the physical behavior, when the volume of copper and silver nanoparticles increases the thermal conductivity increases, and then the thermal boundary layer thickness increases. Moreover Fig. 7 displays the behavior of the different types of nanoparticles on temperature distribution whenϕ=0.1. The figure showed that by using different types of nanofluid as the values of the temperature change and the Ag-nanoparticles are the lower distribution.

Figure 5.

Effects of the nanoparticle volume fractionϕon temperature distributionθ(η)in the case of Cu-Water.

Figure 6.

Effects of the nanoparticle volume fractionϕon temperature distributionθ(η)in the case of Ag-Water.

Figure 7.

Temperature profilesθ(η)for different types of nanofluids whenϕ=0.1.

Fig. 8 shows the variation of the reduced Nusselt number with the nanoparticles volume fraction ϕfor the selected types of the nanoparticles. It is clear that the heat transfer rates decrease with the increase in the nanoparticles volume fractionϕ. The change in the reduced Nusselt number is found to be lower for higher values of the parameterϕ. It is observed that the reduced Nusselt number is higher in the case of Ag-nanoparticles and next Cu-nanoparticles, TiO2-nanoparticles and Al2O3-nanoparticles. Also, the Fig. 8 and Table 3 show that the values of θ(0)change with nanofluid changes, namely we can say that the shear stress and heat transfer rate change by taking different types of nanofluid. Furthermore this depicts that the nanofluids will be very important materials in the heating and cooling processes.

Figure 8.

Effects of the nanoparticle volume fractionϕon dimensionless heat transfer rates.

4. Conclusions

The problem of the steady free convection boundary layer flow past a vertical cone embedded in porous medium filled with a non-Newtonian nanofluid has been studied and the special case when the base fluid is water has been considered. The effects of the solid volume fraction ϕon the flow and heat transfer characteristics are determined for four types of nanofluids: Copper (Cu), Silver (Ag), Alumina (Al2O3) and Titanium oxide (TiO2). It has been shown, as expected, that increasing of the values of the nanoparticles volume fraction lead to an increase of the velocity and the temperature profiles and to an decrease of the Nusselt number for the values of the parameterϕ. It has been found that the Ag-nanoparticles proved to have the highest cooling performance and Alumina-nanoparticles enhanced to have highest heating performance for this problem.


Cpspecific heat at constant temperature

fdimensionless stream function

gacceleration due to gravity

hlocal heat transfer coefficient

Kpermeability coefficient of the porous medium

kthermal conductivity

mgeometry shape parameter

Nuxreduced Nusselt number

nviscosity index,n0

Raxmodified Rayleigh number

rlocal radius of the cone


Twtemperature at the surface of the cone

Tambient temperature attained as y

u,vDarcian velocity components in x- andy-directions

x,yCartesian coordinates

Greek symbols

αthermal diffusivity

βvolumetric expansion coefficient

γhalf angle of the cone

ηsimilarity variable

θdimensionless temperature

μeffective viscosity

ρfdensity of the fluid

ρsnanoparticles mass density

(ρCp)nfheat capacitance of the nanofluid

(ρCp)fheat capacity of the fluid

(ρCp)seffective heat capacity of the nanoparticles material

ϕnanoparticles volume fraction

ψstream function


ffluid fraction

nfnanofluid fraction

ssolid fraction

wcondition at the wall

stream function condition at the infinity

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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F.M. Hady, F.S. Ibrahim, S.M. Abdel-Gaied and M.R. Eid (October 31st 2012). Boundary-Layer Flow in a Porous Medium of a Nanofluid Past a Vertical Cone, An Overview of Heat Transfer Phenomena, Salim N. Kazi, IntechOpen, DOI: 10.5772/53497. Available from:

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